Physical Sciences and Mathematics Commons^™

Upper Bounds For The Isolation Number Of A Matrix Over Semirings, Leroy B. Beasley, Seok-Zun Song

Mathematics and Statistics Faculty Publications

Let S be an antinegative semiring. The rank of an m×n matrix B over S is the minimal integer r such that B is a product of an m×r matrix and an r×n matrix. The isolation number of B is the maximal number of nonzero entries in the matrix such that no two entries are in the same column, in the same row, and in a submatrix of B of the form [b_i,j b_k,j

b_i,l b_k,l] with nonzero entries. We know that the isolation number of B is …

Possible Isolation Number Of A Matrix Over Nonnegative Integers, Leroy B. Beasley, Young Bae Jun, Seok-Zun Song

Mathematics and Statistics Faculty Publications

Let ℤ₊ be the semiring of all nonnegative integers and A an m × n matrix over ℤ₊. The rank of A is the smallest k such that A can be factored as an m × k matrix times a k×n matrix. The isolation number of A is the maximum number of nonzero entries in A such that no two are in any row or any column, and no two are in a 2 × 2 submatrix of all nonzero entries. We have that the isolation number of A is a lower bound of the rank of …